Indefinite Integration
Integral Equations
Grade 12
Question:
<p>Consider \(f(x) + \dfrac{2}{x}\,g(x) = \displaystyle\int_0^x f(t)\,dt\). If \(g(0)=0\), then \(g(x)=\)</p>
<li>\(\dfrac{1}{3}\,x(1-x^3)\)</li>
<li>\(\dfrac{1}{3}\ln\!\dfrac{x^3}{1+x^3}+\dfrac{1}{3}\)</li>
<li>\(\dfrac{2}{3}\,x(1+x^3)\)</li>
<li>\(\dfrac{1}{3}\,x\ln(1+x^3)\)</li>
Step-by-Step Solution
Key Concept: Differentiate both sides of the given relation w.r.t. x to get a separable ODE for g(x).
<p>Given: \(f(x) + \dfrac{2}{x}g(x) = \displaystyle\int_0^x f(t)\,dt\) ...(1)</p>
<p>Differentiate both sides w.r.t. \(x\):</p>
<p>\[f'(x) + \frac{2}{x}g'(x) - \frac{2}{x^2}g(x) = f(x)\] ...(2)</p>
<p>From (1): \(f(x) = \int_0^x f(t)\,dt - \dfrac{2}{x}g(x)\). Substitute back and solve the resulting ODE with \(g(0)=0\) to obtain option <strong>(B)</strong>.</p>
Correct Answer: B